Let's solve the equation:

\[
-3x \times -(7x + 4) = -4(3x + 1)
\]

### Step 1: Simplify both sides.

#### Left-hand side:
\[
-3x \times -(7x + 4) = 3x(7x + 4)
\]
Now, distribute \(3x\) on the left-hand side:
\[
3x(7x) + 3x(4) = 21x^2 + 12x
\]

#### Right-hand side:
Distribute \(-4\) on the right-hand side:
\[
-4(3x + 1) = -12x - 4
\]

Now we have the simplified equation:

\[
21x^2 + 12x = -12x - 4
\]

### Step 2: Bring all terms to one side.

Move all terms to the left-hand side:
\[
21x^2 + 12x + 12x + 4 = 0
\]
Combine like terms:
\[
21x^2 + 24x + 4 = 0
\]

### Step 3: Solve the quadratic equation.

The equation is now:
\[
21x^2 + 24x + 4 = 0
\]

This is a standard quadratic equation, which can be solved using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Where \(a = 21\), \(b = 24\), and \(c = 4\).

### Step 4: Apply the quadratic formula.

First, calculate the discriminant:
\[
b^2 - 4ac = 24^2 - 4(21)(4) = 576 - 336 = 240
\]
Now, substitute into the quadratic formula:
\[
x = \frac{-24 \pm \sqrt{240}}{2(21)} = \frac{-24 \pm \sqrt{240}}{42}
\]

### Step 5: Simplify the solution.

The square root of 240 can be simplified:
\[
\sqrt{240} = \sqrt{16 \times 15} = 4\sqrt{15}
\]
So the solution becomes:
\[
x = \frac{-24 \pm 4\sqrt{15}}{42}
\]
Simplify further:
\[
x = \frac{-24}{42} \pm \frac{4\sqrt{15}}{42}
\]
\[
x = \frac{-4}{7} \pm \frac{\sqrt{15}}{10.5}
\]

Thus, the solutions are:
\[
x_1 = \frac{-4}{7} + \frac{\sqrt{15}}{10.5}
\quad \text{and} \quad
x_2 = \frac{-4}{7} - \frac{\sqrt{15}}{10.5}
\]

Would you like more details on this solution? Let me know if you have any questions.

Here are 5 related questions:
1. How does the quadratic formula help solve any quadratic equation?
2. What is the importance of simplifying square roots in algebraic solutions?
3. Why do we move all terms to one side in a quadratic equation?
4. Can all quadratic equations be factored, or do some require the quadratic formula?
5. What happens when the discriminant is negative in a quadratic equation?

**Tip:** Always check the discriminant (\(b^2 - 4ac\)) before solving a quadratic equation; it tells you if there are real, complex, or repeated solutions.

This page provides a detailed solution to the quadratic equation -3x × -(7x + 4) = -4(3x + 1), involving algebraic simplification, distribution, and solving using the quadratic formula. The solution is suited for students in Grades 9-11.

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